Gas-phase Modeling of the Cometary Coma of Interstellar Comet 2I/Borisov

The initial gas velocity at the nuclear surface is assumed to be equal to the local sonic speed (Weiler 2006) given by

$$\begin{eqnarray}&&{v}_{0}=\sqrt{\gamma {R}_{g}{T}_{0}},\end{eqnarray} \tag{ 1 }$$

where γ is the adiabatic exponent, R_g is the specific gas constant, and T₀ is the initial temperature of the coma gas. This temperature can be calculated from the surface ice temperature T_s . A Maxwellian velocity distribution of the outflowing gas is reached only after several molecular collisions, after which T₀ and T_s are related as:

$$\begin{eqnarray}&&{T}_{0}=\displaystyle \frac{{T}_{s}}{1+\tfrac{1}{2}(\gamma -1)}.\end{eqnarray} \tag{ 2 }$$

The initial temperature of all of the three fluids is set in this manner. In order to find the surface temperature, the energy balance equation is solved, and for a pure icy surface, this equation can be written as:

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{\odot }(1-{A}_{v})}{{r}_{h}^{2}}\left\langle \cos \theta \cos \phi \right\rangle ={\epsilon }_{\mathrm{IR}}{\sigma }_{B}{T}_{s}^{4}+{HZ}(T),\end{eqnarray} \tag{ 3 }$$

where F_⊙ is the incident solar flux, which is scaled by the heliocentric distance r_h , A_v is the visual albedo, θ and ϕ are the local hour angle and the latitude, _IR is the infrared emissivity, σ_B is the Stefan−Boltzmann constant, H is the latent heat of sublimation of the ice, and Z(T) is the surface sublimation rate. If Q_0,pri is the production rate of a parent volatile from the nucleus and r₀ is the nuclear radius, then the number density of that volatile near the surface is

$$\begin{eqnarray}&&{n}_{0,\mathrm{pri}}=\displaystyle \frac{{Q}_{0,\mathrm{pri}}}{4\pi {r}_{0}^{2}{v}_{0}}.\end{eqnarray} \tag{ 4 }$$

To study the molecular complexity observed in the cometary coma, one needs chemical networks that contain reaction rate constants. The rate coefficients of bimolecular processes are expressed in the form of a so-called modified Arrhenius formula as follows:

$$\begin{eqnarray}&&{k}_{j}={\alpha }_{j}{\left(\displaystyle \frac{T}{300\ {\rm{K}}}\right)}^{{\beta }_{j}}{e}^{-{\gamma }_{j}/T}\end{eqnarray} \tag{ 5 }$$

where α_j , β_j , and γ_j are parameters obtained from literature and databases. For some reactions, different values of the parameters are needed for different temperature ranges. Rate coefficients for unmeasured ion−neutral reactions are calculated by Woon & Herbst (2009) using the Su−Chesnavich capture approach. Two separate formulae are used to compute the rate coefficients, one for lower temperatures and the other for higher temperatures, given by Equations (6) and (7), respectively.

$$\begin{eqnarray}&&k(T)=\alpha \beta \left[0.62+0.4767\gamma {\left(\displaystyle \frac{300\ {\rm{K}}}{T}\right)}^{0.5}\right].\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&k(T)=\alpha \beta \left[1+0.0967\gamma {\left(\displaystyle \frac{300\ {\rm{K}}}{T}\right)}^{0.5}+\displaystyle \frac{{\gamma }^{2}}{10.526}\displaystyle \frac{300\ {\rm{K}}}{T}\right].\end{eqnarray} \tag{ 7 }$$

The chemical reactions used in the model are taken from Schmidt et al. (1988), Weiler (2006), and Hölscher (2015), with necessary updates from the KIDA database (Wakelam et al. 2015, http://kida.astrophy.u-bordeaux.fr/). It also contains the parameters required to calculate the rate coefficients for the bimolecular reactions using Equation (5) and the ion−neutral reactions using Equations (6) and (7). The chemical network we have used contains more than 400 chemical species connected by about 4500 reactions. The reactions due to the electron impact excitation and radiative de-excitation of the electronic states of CO are taken from Schmidt et al. (1988). Reactions involving the metastable excited states of oxygen, carbon, and nitrogen are taken from Raghuram & Bhardwaj (2013) and Raghuram et al. (2020). The photolytic reaction rates are taken from Huebner & Mukherjee (2015) and Weiler (2006), appropriate for a quiet Sun. Optical depth effects are present in the coma, because of which the rates of the photochemical reactions decrease in the innermost regions. A detailed calculation of the photorates in the inner regions is done by dividing the solar UV radiation flux into various wavelength bins. The optical depth in each bin is then calculated using the photo cross sections and species number densities (details in A.5 of the Appendix). The binned photo cross sections are available at https://phidrates.space.swri.edu/#. The wavelength-dependent solar irradiance data set is available with the LASP Interactive Solar Irradiance Data Center (LISIRD), and we have used the flux derived from the FISM2 (0.1–190 nm) and NRLSSI2 (>190 nm) models (https://lasp.colorado.edu/lisird/). The flux is scaled to the heliocentric distance r_h of the comet as ${r}_{h}^{-2}$.

Figure 1 shows the production rates for observed volatiles for comet 2I; some of these are parent volatiles sublimating directly from the nucleus, while some are products of photochemical reactions. The molecular production rates for the different volatiles that are used as the initial inputs in our gas-phase model are given in Table 1. We modeled the coma at a heliocentric distance of 2.01 au post-perihelion; 2I was at this distance on 2019 December 15–16 and Cordiner et al. (2020) report on the production rates of several volatiles derived from observations made on these dates. The production rates of some of the cometary volatiles observed in comet 2I are not available for these dates. Hence, for these volatiles, we used the production rates derived from observations made on other dates or estimated them from some fitted relation. Cordiner et al. (2020) report Q_CO = (4.4 ± 0.7) × 10²⁶ mol s⁻¹ from the interferometric spectra and Q_CO = (5.0 ± 0.5) × 10²⁶ mol s⁻¹ using the autocorrelation (total power) spectra. Bodewits et al. (2020) report Q_CO = (6.4 ± 1.4) × 10²⁶ mol s⁻¹ for the observational epoch 2019 December 19–22. We use a conservative value of Q_CO = 5.0 × 10²⁶ mol s⁻¹, which is within the measured error limits. Bodewits et al. (2020) derived an empirical fitting for the H₂O production rate. We have used this empirical relation to estimate ${{\rm{Q}}}_{{{\rm{H}}}_{2}{\rm{O}}}$ around 2019 December 15.

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Cordiner et al. (2020) obtained the production rate of HCN and also derived upper limits on the production rates of CS and CH₃OH. The upper limits on the relative abundances (with respect to H₂O) of these species are consistent with values previously observed in solar system comets. We used the upper limit for CS derived by Cordiner et al. (2020) as our initial abundance for this species. The initial production rate of CH₃OH is taken to be about 2% of H₂O. This is a conservative value, which is roughly the average abundance of CH₃OH with respect to H₂O in solar system comets (Dello Russo et al. 2016). H₂CO is an intermediate species in the formation of CH₃OH by successive hydrogenation of CO. Thus H₂CO is also expected to be present in the coma, although it is not detected. We obtained additional model runs with H₂CO as a parent species. Lin et al. (2020) derived a power-law variation with heliocentric distance for the C₂ production rate. Using this relation, and their measured ${{\rm{Q}}}_{{{\rm{C}}}_{2}}=(5.5\pm 0.4)\times {10}^{23}$ mol s⁻¹ at 2.145 au, we calculated ${{\rm{Q}}}_{{{\rm{C}}}_{2}}$ at 2.01 au for our model run. The production rate for NH₂ was estimated by Bannister et al. (2020) from observations made in late 2020 November at a distance of 2.03 au pre-perihelion.

CS, C₂, and NH₂ are product species that are formed in the coma directly or indirectly by photodissociation of parent species. The CS radical is most likely to trace the parent molecule CS₂ (Bockelée-Morvan et al. 2004). C₂ is produced from several sources, including as a granddaughter species from the photodissociation of C₂H₂ (Combi & Fink 1997). The most likely candidate for the formation of NH₂ in comets is the short-lived parent species NH₃, and the abundance of NH₂ is a direct measurement of the NH₃ abundance (Wyckoff et al. 1989). We have converted the abundances of CS, C₂, and NH₂ into their parent species abundances from the branching ratios of the photolytic reactions (Fink & Disanti 1990).

Lee et al. (2020) used infrared imaging to infer an upper limit of 0.58 km for the nuclear radius r_n . Jewitt et al. (2020) constrained the radius to be ≤0.5 km from measurements of the surface brightness profiles from the Hubble Space Telescope and an assumed albedo of 0.04. Guzik et al. (2020) estimated r_n ∼ 1 km from the observed water production rate and an active fraction of 30%. We assumed a value of 1 km as the radius of the nucleus.

The FISM2 and NRLSSI2 flux models give the daily solar spectral irradiances. We collected the daily fluxes for the entire month of 2019 December from the LISIRD database. We then found the average spectral fluxes for 2019 December and scaled them to 2.01 au heliocentric distance. We used these scaled fluxes to derive the photorates as described in the Appendix A.5.

The temperature profiles for the cases (1) and (2) mentioned above are shown in Figure 2, and Figure 3 shows some of the major heating/cooling rates per unit volume for the neutral and electron fluid. Our temperature profile for the Halley-type comet is qualitatively similar to the temperature profile for a Hyakutake-like comet obtained by Weiler (2006). The neutral temperature profiles shown in Figures 2(a) and 2(b) are relatively similar, with an initial dip and then a temperature inversion at a cometocentric distance of ∼200 km. However, the neutral fluid temperature falls to a much lower value (∼6 K) near inversion for comet 2I, as compared to an inversion temperature of ∼18 K for a Halley-type comet. In fact, at any cometocentric distance, the neutral temperature of the coma of the solar system comet is more than that of the interstellar comet. The main sources of heat for the neutral fluid in the coma are photodissociation and exothermic reactions. The photodissociation of H₂O into OH and H produces 3.41 eV of energy, while CO photodissociates into C and O to produce 2.56 eV of energy (Huebner & Mukherjee 2015). Thus, the net rate of heat energy released per unit volume due to photodissociation is higher in the Halley-type comet because of a greater relative abundance of H₂O. Besides, due to the larger production rate and higher density of the coma of the Halley-type comet, the net heating rate of the neutral fluid per unit volume from chemical reactions is greater.

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The initial coupling between the electron and neutral temperatures is relatively weak for the comet 2I and they decouple at a shorter distance (∼20 km). The coupling is due to the inelastic collisions between the electrons and the water molecules, leading to rotational excitation of water molecules. The energy exchange rate due to collisions depends on the collision cross section as well as the number densities of the colliding species. The lower coma density for comet 2I implies that the frequency of such collisions is reduced. In addition, the dominant neutral species in the coma of comet Halley is H₂O, while 2I has nearly equal production rates of both H₂O and CO. This means that the relative number of H₂O molecules available for rotational excitation is lesser in the interstellar comet. Therefore, the rate of energy exchange between the neutral and electron fluids is reduced in the innermost coma regions of 2I when compared to that of a Halley-type comet, as can be seen from Figure 3. The electron fluid loses less of its energy to the neutral fluid in comet 2I, and its temperature decouples closer to the nucleus. In the outer part of the coma, the electron temperature is high and vibrational excitation of neutral species by electrons becomes an important cooling mechanism for electrons. It is seen from Figure 3 that cooling of electrons due to vibrational excitation for CO is comparable to or larger than that for water for the coma of comet 2I. The energy loss rate per unit volume for electrons due to elastic scattering from the neutral species is several orders of magnitude less than the loss due to inelastic collisions (Figure 3).

Comet 2I is unique in the sense that it contains H₂O and CO in nearly equal proportions, which will have a considerable bearing on the ionic abundances. The variation of the flux of parent volatiles, electrons, and assorted ionic species as a function of cometocentric distance is shown in Figure 4. The flux is calculated as 4π r² vn, where v is the velocity and n is the species number density. Plotting the flux removes the r⁻² dependence, and rising/falling curve indicates that the species is getting created/destroyed by chemical reactions. H₂O and CO together account for nearly 99% of the outgassing from the nucleus. Most of the parent species survive up to 10⁵ km, consistent with an outflow velocity of ∼1 km s⁻¹ and an average photolytic reaction rate of ≳ 10⁻⁵ s⁻¹, leading to a nearly constant flux throughout the modeled region of the coma. NH₃ and CS₂ are exceptions, as they are destroyed faster by photolytic processes. We also note that since the number density is low, optical depth effects are nearly absent in the coma. Thus, the rates for the photoprocesses remain nearly constant throughout the coma.

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The ions H₃O⁺, HCO⁺, NH₄ ⁺, HCNH⁺, and CH₃OH⁺ that are formed by the protonation (transfer of H⁺ ion) of parent species show an initial steep rise in the flux. As the density decreases on moving outwards, the timescale for the formation of these species becomes more than the dynamical timescale, and the curves tend to flatten. The flux of ions formed by photoionization or photodissociative ionization of parent species keeps rising in the outer coma as well, since the scale length for most of the parent species is ≳10⁵ km. The dominant formation/destruction pathways for assorted ions, along with their relative reaction rates, are shown in Figures 5–7. The relative reaction rate is calculated by dividing the reaction rate for the creation/destruction process of a species by the total creation/destruction rate for that species. The relative reaction rate helps in understanding the contribution of each reaction toward the net creation or destruction of a species in different regions of the coma.

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4.2.1. CO⁺ and HCO⁺

4.2.2. H₂O⁺ and H₃O⁺

The variation of the flux of H₂O⁺ and H₃O⁺ is shown in Figure 4(b) and their major formation and destruction pathways in Figure 6. The H₂O⁺ ion, in addition to being produced by photoionization of H₂O, is also produced when H₂O participates in charge exchange reactions with ions present in the coma. CO⁺ is the most abundant ion with which H₂O undergoes charge exchange, and this reaction accounts for 30%–35% of the net production of H₂O⁺ within 100 km. H₂O⁺ is mainly destroyed by proton transfer reactions with H₂O and CO, and also with OH beyond ∼500 km. As with CO⁺ and HCO⁺, dissociative recombination reactions become steadily more important, and their contribution to the loss processes can rise up to 20%–25% in the outer regions.

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H₃O⁺ is one of the more abundant ions in the coma, and it forms mostly by protonation of H₂O by ions such as H₂O⁺, HCO⁺, and OH⁺. Proton transfer reactions with the parent species CH₃OH, NH₃, and HCN account for most of the loss of H₃O⁺ within ∼10³ km. The relative rate for destruction of H₃O⁺ by dissociative recombination is much more than that for other ions such as H₂O⁺, HCO⁺, and CO⁺. In the outer regions beyond 10⁴ km, this is the most efficient loss mechanism for H₃O⁺.

4.2.3. N-bearing Ions

The variation of flux as a function of cometocentric distance for major N-bearing ionic species is shown in Figure 4(c) and the reaction pathways in Figure 7. Most of the ${\mathrm{NH}}_{2}^{+}$ in the coma is formed by photodissociative ionization of NH₃, and a very small amount (∼2%) by charge exchange reactions between NH₂ and abundant ions. ${\mathrm{NH}}_{3}^{+}$ is formed by photoionization of NH₃ and charge exchange of NH₃ with ions (mainly CO⁺ and H₂O⁺). Photoionization and charge exchange contribute nearly equally to ${\mathrm{NH}}_{3}^{+}$ ion formation near the nucleus, but beyond 50 km, the charge exchange reaction rates begin to fall. The destruction of both ${\mathrm{NH}}_{2}^{+}$ and ${\mathrm{NH}}_{3}^{+}$ is primarily caused by H₂O molecules. ${\mathrm{NH}}_{2}^{+}$ reacts with H₂O to form the protonated ion H₃O⁺, and the relative rate for this reaction is 90%. ${\mathrm{NH}}_{2}^{+}$ also reacts with H₂O at slower rates to form ${\mathrm{NH}}_{3}^{+}$ and NH₄ ⁺ ions. ${\mathrm{NH}}_{3}^{+}$ reacts with H₂O, and also with NH₃ and OH to form NH₄ ⁺. Similar to previously discussed ions, dissociative recombination rates of ${\mathrm{NH}}_{2}^{+}$ and ${\mathrm{NH}}_{3}^{+}$ increase on moving outwards in the coma.

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NH₃ undergoes proton transfer reaction with many of the ions such as H₃O⁺, H₂O⁺, and HCO⁺ to form NH₄ ⁺. Apart from this, neutral species (mainly H₂O and NH₃) undergo bimolecular reaction with ${\mathrm{NH}}_{3}^{+}$, and form NH₄ ⁺ by breaking the hydrogen bond. NH₄ ⁺ is mainly destroyed by dissociative recombination that proceeds via three channels.

HCN⁺ forms by photoionization of the parent molecule HCN and is destroyed by proton transfer and charge exchange reactions with the parent species CO and H₂O. HCN undergoes protonation by ions such as H₃O⁺, H₂O⁺, and HCO⁺ to form HCNH⁺, and HCNH⁺ then successively causes protonation of neutral molecules H₂O, NH₃, and H₂CO.

4.2.4. Complex Ions

The electrons that are created due to photoionization and photodissociative ionization processes gain most of the excess energy released from the corresponding photochemical reaction because of their lighter mass. A dominant heat loss mechanism for electrons is inelastic scattering from neutral species, leading to the excitation of neutrals to the higher rotational and vibrational energy levels. H₂O is a highly polar molecule and, in a water-dominated coma, inelastic scattering from this molecule is the primary reason for cooling of the electron fluid. Previous hydrodynamical models have only considered electron cooling due to water molecules because of a lesser abundance of other neutral species and a high e⁻−H₂O collision cross section. In the case of the interstellar comet, where the CO abundance can be as high as or even higher than the H₂O abundance, e⁻−CO inelastic collisions cannot be ignored. We have shown that in the high electron temperature and lower density region of the coma, e⁻−CO vibrational collisions are an important cooling mechanism. Electron-neutral temperature coupling exists up to some distance from the nucleus because the energetic photoelectrons efficiently lose their energy to the neutral fluid by rotationally exciting water molecules. This coupling that is observed in a water-dominated coma is also seen in the interstellar comet, though the coupling is weaker because of the reduced density, resulting from a lower production rate. The presence of a large amount of CO also reduces the net relative abundance of H₂O, thereby reducing the cooling efficiency. If the CO abundance is increased even more, e.g., in comet C/2016 R2 (Biver et al. 2018; Wierzchos & Womack 2018; McKay et al. 2019), coupling may be lost altogether. However, it should be kept in mind that we have not considered rotational excitation of CO resulting from e⁻−CO inelastic collisions. Since the e⁻−CO collision cross section for rotational excitation is nearly four orders of magnitude less than that of e⁻−H₂O collision, the net effect of including rotational excitation of CO may cause only a fractional change in the temperature profile of comet 2I.

Many types of chemical reactions occur in the coma, but two particularly important processes that drive ion chemistry in the coma in terms of creating new ions are photochemical reactions and proton transfer reactions. The dominant ions in the coma are the protonated parent species and the products of photoionization and photodissociative ionization of the parent molecules. The protonated molecules are more abundant within ∼100 km, while the photoproduct ion abundance takes over in the outer regions. The abundant ions H₃O⁺ and HCO⁺, formed by protonation of H₂O and CO, in turn cause protonation of other parent species to produce other ions, namely NH₄ ⁺, HCNH⁺, and ${\mathrm{CH}}_{3}{\mathrm{OH}}_{2}^{+}$. Hence, an important feature of the coma chemistry is a protonation"chain" whereby successive protonated species are created.

Ions in the coma undergo dissociative recombination reactions, which are dependent on the temperature of the electrons. The rates for these reactions are determined by Equation (5), where T is the electron temperature and, typically, the parameters β_j ∼ −0.5 and γ_j = 0. Thus, these reaction rates decrease with an increase in the electron temperature and vice versa. From Figures 5–7, we see that all of the dissociative recombination reaction rates show an inversion at ∼30 km, and the relative loss of ions by these processes slows down sharply. This happens because there is an electron temperature inversion, followed by a sudden increase in the electron temperature at this distance. The primary mechanism by which electrons lose heat in the coma is due to inelastic collisions with the neutral species. The electron temperature decouples from the neutral temperature once the frequency of such collisions is reduced (Section 4.1). Subsequently, the heating rate of the electrons per unit volume due to chemical reactions becomes more than the cooling rate, resulting in the temperature inversion. On moving outward in the coma, the relative rates of the dissociative recombination reactions again increase, due to the continuous increase in the flux of photoelectrons. This inversion in the dissociative recombination rate is also seen in the Halley-type comet, though it occurs further out from the nucleus, at a distance of ∼1300 km, due to the delay in the electron fluid decoupling and temperature inversion. Besides, the relative reaction rates reduce at most by a factor of 2 in the Halley-type comet, whereas they can reduce by up to a factor of 4 in the interstellar comet. The reduction in the rate coefficients is compensated for by the enhanced electron flux, resulting in a lower net reduction for the Halley-type comet. The rates at which the chemical reactions occur in the coma depend on a host of factors such as the temperature dependence of the reaction rates, the availability of the reacting species and the optical depth effects (for photochemical reactions). In general, we see that the relative rates for the reactions involved in forming or destroying a particular species do not necessarily stay constant throughout. The dominant process by which a species is created or destroyed gets altered in different regions of the coma.

The production of H₂O⁺ ion generally occurs by photoionization of H₂O in water-dominated comets. However for comets with a large amount of CO like 2I/Borisov, a large amount of CO⁺ will be produced and charge exchange of H₂O with CO⁺ is also an important formation mechanism of H₂O⁺. A large amount of HCO⁺ is also present in our modeled coma of comet 2I because of a high CO and CO⁺ abundance. Such high quantities of HCO⁺ would not be seen in water-dominated comets because of a lesser amount of CO. HCO⁺ is an important ion that causes protonation of neutral species. The approximate contribution of HCO⁺ to the formation rate of H₃O⁺, NH₄ ⁺, and ${\mathrm{CH}}_{3}{\mathrm{OH}}_{2}^{+}$ is up to 30%–40%, ∼5%, and up to 20%, respectively.

The flux profiles of H₂O⁺ and CO⁺ ions for comet 2I are nearly alike and this can be explained as follows. These two ions are primarily produced by photoionization at similar rates. In terms of loss processes, H₂O⁺ is destroyed mainly by proton transfer reactions with H₂O and CO, while CO⁺ is destroyed by charge exchange and ion−neutral reactions with H₂O. The Arrhenius rate coefficients (Equation (5)) for the loss processes for H₂O⁺ and CO⁺ are similar. The production and loss rates will also depend on the number densities of H₂O and CO, which again, are almost alike.

The presence of CH₃OH is not conclusively found in the comet 2I; Cordiner et al. (2020) reported an upper limit of 4.4 × 10²⁶ mol s⁻¹ on the production rate of CH₃OH. Solid state laboratory studies show that CH₃OH can form in CO-rich ice at low temperature, by successive hydrogenation of CO, while H₂CO forms as an intermediate product (Hiraoka et al. 1994; Watanabe & Kouchi 2002; Fuchs et al. 2009). The large amount of CO in the icy nucleus of comet 2I can produce sufficient quantities of CH₃OH, but it was not conclusively detected, possibly due to lower activity. Since 2I shows similarity to other solar system comets in terms of composition of some of the volatiles, we have used the average value of CH₃OH for solar system comets (Dello Russo et al. 2016) as the production rate in our model. We have not included other organic species such as CH₄ and C₂H₆ as parent species in our model for comet 2I, as there is no report on their detection, though these species are generally observed in solar system comets.

To see which of the organic species abundances are affected when we have more CO outgassing from the nucleus, we ran the model for the compositions given in Tables 1 and 2 but for a higher production rate of ∼10²⁹ mol s⁻¹. Since hydrogenation of CO will form H₂CO as an intermediate product, we also ran an additional model for the 2I-type comet composition by adding a relative H₂CO abundance that is 4% of that of water, which is also the H₂CO abundance in the Halley-type comet composition. Figure 8 shows the flux variation for some of the larger organic ions and neutrals for the 2I-type and Halley-type cometary compositions. The species shown in Figure 8(a) exhibit a higher abundance in the coma for the 2I-type comet as compared to the Halley-type comet, by ∼1 order of magnitude. This can be understood from the reaction pathways discussed in Section 4.2.4. ${\mathrm{CH}}_{3}{\mathrm{OH}}_{2}^{+}$ forms when H₃O⁺ and HCO⁺ ions cause protonation of CH₃OH. There is a considerably higher abundance of HCO⁺ ions in the coma of the 2I-type comet as opposed to the Halley-type comet, which increases the net protonation reaction rate of CH₃OH and results in a higher ${\mathrm{CH}}_{3}{\mathrm{OH}}_{2}^{+}$ abundance. An increased density of ${\mathrm{CH}}_{3}{\mathrm{OH}}_{2}^{+}$ will result in a higher abundance of ${\mathrm{CH}}_{3}{\mathrm{OCH}}_{4}^{+}$ (formed by methyl cation transfer from ${\mathrm{CH}}_{3}{\mathrm{OH}}_{2}^{+}$ to CH₃OH). Dissociative recombination of ${\mathrm{CH}}_{3}{\mathrm{OCH}}_{4}^{+}$ then leads to the formation of CH₃OCH₃. The radiative association of H₂O and HCO⁺ leads to the formation of ${\mathrm{CH}}_{3}{{\rm{O}}}_{2}^{+}$, and this ion then undergoes dissociative recombination to form HCOOH. We can see that a high HCO⁺ abundance (resulting from a high CO abundance) is key to the increased abundance of these species in a 2I-type comet composition. Thus, the presence of large quantities of CO will accelerate the formation of some of the organic ions and neutrals in the gas phase.

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CH₃OH undergoes photoionization to form CH₃OH⁺ and neutral−neutral collision reactions to form CH₂OH and CH₃O. Since there is a nearly equal abundance of CH₃OH in the 2I-type and Halley-type comet compositions, these product species also show near similar abundances, as seen in Figure 8(b). CH₃O and CH₂OH show a sharp fall in the outermost region as they are destroyed by H₂O in this region. The energy barrier for the destruction reactions is high, and this barrier is crossed in the high-temperature region of the outer coma.

Figures 9(a) and (b) show the abundance of assorted species with and without H₂CO added to the 2I-type comet composition. H₂CO⁺ forms by photoionization of H₂CO and photodissociative ionization of CH₃OH, while protonation of H₂CO produces H₂COH⁺. H₂CO also undergoes an ion−neutral reaction with ${\mathrm{CH}}_{3}{\mathrm{OH}}_{2}^{+}$ to form ${{\rm{H}}}_{5}{{\rm{C}}}_{2}{{\rm{O}}}_{2}^{+}$ (protonated methyl formate), which successively forms HCOOCH₃ by dissociative recombination. Thus in the presence of H₂CO as a parent volatile, these species are seen in larger quantities in both the Halley-type and 2I-type comet compositions, as is evident from Figure 9(a). Besides, the abundance is higher in the 2I-type composition, as compared to the Halley-type composition. This is because of a higher availability of HCO⁺ and ${\mathrm{CH}}_{3}{\mathrm{OH}}_{2}^{+}$ ions in the 2I-type comet composition. HCO⁺ causes protonation of H₂CO to form H₂COH⁺, and ${\mathrm{CH}}_{3}{\mathrm{OH}}_{2}^{+}$ reacts with H₂CO to form successive species, as discussed above. When H₂CO is not considered as a parent species for the comet 2I-type composition, only a small amount of H₂CO is formed in the coma by the photodissociation of CH₃OH. This is not sufficient to produce a higher abundance of molecules such as HCOOCH₃ as can be seen from Figure 9(b). Thus, we can say that if any organic molecule is present as a parent species in a comet, then the ions and neutrals produced from this molecule will have a high abundance in the coma. In addition, the abundance of these product species will be more in a comet where there is enhanced CO outgassing.

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The governing equations for the steady state, spherically symmetric gas-phase coma (Weiler 2006, 2012; Hölscher 2015) are as follows:

$$\begin{eqnarray}&&\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{d}{{dr}}({r}^{2}{nv})=N,\end{eqnarray} \tag{ A1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{d}{{dr}}({r}^{2}\rho v)=M,\end{eqnarray} \tag{ A2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{d}{{dr}}({r}^{2}\rho {v}^{2})+\displaystyle \frac{d}{{dr}}({{nk}}_{B}T)=F,\end{eqnarray} \tag{ A3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{d}{{dr}}\left[{r}^{2}\rho v\left(\displaystyle \frac{{v}^{2}}{2}+\displaystyle \frac{\gamma }{\gamma -1}\displaystyle \frac{{k}_{B}T}{\mu }\right)\right]=Q.\end{eqnarray} \tag{ A4 }$$

These equations represent the conservation of number density, mass, momentum, and energy, respectively, for each fluid. In the above equations, r is the cometocentric distance, n is the number density of any chemical species, v is the common velocity for all fluids, and k_B is the Boltzmann constant. Variables ρ, T, γ, and μ are, respectively, the mass density, temperature, adiabatic exponent, and average molecular mass corresponding to each fluid. The source terms N, M, F, and Q represent the net rate per unit volume for the generation of chemical species, mass, momentum, and energy, respectively. The energy source term Q can be replaced by the thermal energy source term G, such that

$$\begin{eqnarray}&&G=Q-{Fv}+\displaystyle \frac{1}{2}{{Mv}}^{2}.\end{eqnarray} \tag{ A5 }$$

The Equations (A1) to (A4) are rewritten as a set of coupled first-order differential equations in n, v, and T, as follows.

$$\begin{eqnarray}&&\displaystyle \frac{{{dn}}_{j}}{{dr}}=\displaystyle \frac{{N}_{j}}{v}-\displaystyle \frac{{n}_{j}}{v}\displaystyle \frac{{dv}}{{dr}}-\displaystyle \frac{2{n}_{j}}{r},\end{eqnarray} \tag{ A6 }$$

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{dv}}{{dr}}=\sum _{k}\left[\displaystyle \frac{1}{{\rho }_{k}{v}^{2}-{\gamma }_{k}{n}_{k}{k}_{B}{T}_{k}}\right.\\ \quad \left.\times \left({F}_{k}v-({\gamma }_{k}-1){G}_{k}-{M}_{k}{v}^{2}+\displaystyle \frac{2v}{r}{\gamma }_{k}{n}_{k}{k}_{B}{T}_{k}\right)\right],\end{array}\end{eqnarray} \tag{ A7 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{dT}}_{k}}{{dr}}=\displaystyle \frac{({\gamma }_{k}-1){T}_{k}}{v}\left(\displaystyle \frac{{G}_{k}}{{n}_{k}{k}_{B}{T}_{k}}-\displaystyle \frac{2v}{r}-\displaystyle \frac{{dv}}{{dr}}-\displaystyle \frac{{N}_{k}}{({\gamma }_{k}-1){n}_{k}}\right).\end{eqnarray} \tag{ A8 }$$

In the above equations, the subscript j indicates any particular chemical species and the subscript k = n, i, e stands for the three different fluids, namely neutral, ion, and electron, respectively. The terms n_k and N_k are obtained by, respectively, summing over the number densities and number density source terms corresponding to the fluid k. The mass density ρ_k for any fluid is directly calculated from the number density as

$$\begin{eqnarray}&&{\rho }_{k}=\sum _{l}{m}_{l}{n}_{l},\end{eqnarray} \tag{ A9 }$$

where the summation is done over all of the species belonging to the fluid k, and m is the species molecular weight. The adiabatic exponent γ_k for a fluid is calculated as a weighted average of the adiabatic exponents γ for each species belonging to that fluid, with the species number densities as weights:

$$\begin{eqnarray}&&{\gamma }_{k}=\displaystyle \frac{\sum _{l}{\gamma }_{l}{n}_{l}}{\sum _{l}{n}_{l}}.\end{eqnarray} \tag{ A10 }$$

The value of γ for most species is listed in Schmidt et al.(1988).

The chemical reaction network (Section 2.2) contains rate coefficients that vary over many orders of magnitude. As a result, the differential equations used to model the coma are stiff. An implicit/semi-implicit integration method is required to solve the equations. Rosenbrock methods are reliable for solving higher-order stiff systems, and stable solutions can be arrived at by solving the Jacobian matrix. The first practical application of these methods was done by Kaps and Rentrop, so they are also known as Kaps−Rentrop methods. We have adopted the Fortran stiff integrator based on the Kaps−Rentrop algorithm as the numerical scheme (Press et al. 1992, Section 16.6).

The number density source term N_j for a species j is calculated by adding or subtracting together the rate coefficients of the reactions that contribute to the creation or destruction of that species.

$$\begin{eqnarray}&&{N}_{j}=\sum _{j^{\prime} }{\nu }_{j^{\prime} }{{ \mathcal R }}_{j^{\prime} }.\end{eqnarray} \tag{ A11 }$$

Here, ν is the number of molecules of the species created in a particular reaction, ${ \mathcal R }$ is the reaction rate in cm⁻³ s⁻¹, and the summation index $j^{\prime}$ runs over all of those reactions in which the species j is a product or a reactant. Since the different fluids considered in this model have separate temperatures, the temperature-dependent collisional reaction rates are calculated at an effective temperature T_eff (Flower 1985), such that

$$\begin{eqnarray}&&{T}_{\mathrm{eff}}=\displaystyle \frac{{m}_{k}{T}_{l}+{m}_{l}{T}_{k}}{{m}_{l}+{m}_{k}}.\end{eqnarray} \tag{ A12 }$$

T_l and T_k are the fluid temperatures to which the reacting species having masses m_l and m_k belong. For reactions involving collisions with electrons, T_eff ≈ T_e , since the mass of the electron is much less than the mass of the other reacting species.

The total mass and momentum source terms are conserved, that is

$$\begin{eqnarray}&&\sum _{k}{M}_{k}=0,\end{eqnarray} \tag{ A13 }$$

$$\begin{eqnarray}&&\sum _{k}{F}_{k}=0.\end{eqnarray} \tag{ A14 }$$

Since these terms appear only as summations in Equation (A7), they need not be calculated individually for each fluid. The thermal energy source terms G_k for each fluid are sums of different components, which are discussed in the next section.

The thermal energy source terms for the neutral, ion, and electron fluids, respectively, are defined as:

$$\begin{eqnarray}&&{G}_{n}={G}_{n}^{\mathrm{chem}}+{G}_{i,n}^{\mathrm{elas}}+{G}_{e,n}^{\mathrm{elas}}+{G}_{e,n}^{\mathrm{inel}}-{G}_{n}^{\mathrm{rad}},\end{eqnarray} \tag{ A15 }$$

$$\begin{eqnarray}&&{G}_{i}={G}_{i}^{\mathrm{chem}}-{G}_{i,n}^{\mathrm{elas}}-{G}_{e,i}^{\mathrm{elas}},\end{eqnarray} \tag{ A16 }$$

$$\begin{eqnarray}&&{G}_{e}={G}_{e}^{\mathrm{chem}}+{G}_{e,i}^{\mathrm{elas}}-{G}_{e,n}^{\mathrm{elas}}-{G}_{e,n}^{\mathrm{inel}}.\end{eqnarray} \tag{ A17 }$$

These components arise due to chemical reactions, the exchange of energy due to elastic and inelastic scattering processes, and radiative losses.

A.4.1. Chemical Reactions

The components ${G}_{n}^{\mathrm{chem}}$, ${G}_{i}^{\mathrm{chem}}$, and ${G}_{e}^{\mathrm{chem}}$ denote the contribution to the thermal energy source terms of the neutral, ion, and electron fluids, respectively, arising due to chemical reactions. Production or loss of a chemical species results in the addition or removal of the kinetic energy of that species from the fluid that it belongs to. If a species j is created (or destroyed) by the reaction α, then its rate of increase (or decrease) of energy per unit volume is (Draine 1986):

$$\begin{eqnarray}&&{G}_{j,\alpha }={{ \mathcal R }}_{j,\alpha }\left(\tfrac{1}{2}{m}_{j}{\left|{{\boldsymbol{v}}}_{j}-{{\boldsymbol{w}}}_{j,\alpha }\right|}^{2}+\tfrac{1}{2}{m}_{j}{\zeta }_{j,\alpha }^{2}\right).\end{eqnarray} \tag{ A18 }$$

Here, ${{ \mathcal R }}_{j,\alpha }$ is the creation or destruction rate of species j by the reaction α, m_j is the mass, v _j is the mean velocity of species j, w _j,α is the mean velocity of the particles of species j emerging from or being used up by the reaction α, and ζ_j,α is the random velocity of the particles of species j involved in the reaction α. Since the bulk velocity is assumed to be the same for all of the species, ${{\boldsymbol{v}}}_{j}-{{\boldsymbol{w}}}_{j,\alpha }=\vec{0}$, and only the internal energy $\tfrac{1}{2}{m}_{j}{\zeta }_{j,\alpha }^{2}$ is required to compute G_j,α . Chemical reactions also increase or decrease the energy of the system, depending on whether they are exothermic or endothermic, and this is the excess energy ΔE. The fraction of the excess energy received by the products is computed from energy−momentum conservation in the reference frame of the center of momentum of the reactants. Electrons that are created as products receive all of the excess energy, since their mass is negligible in comparison with that of the other species.

The thermal energy source terms for the different reaction types were computed by Draine (1986) and extended by Rodgers & Charnley (2002) and Weiler (2006). Table 3 lists the source terms ${\hat{G}}_{k}$, k = n, i, e for the species belonging, respectively, to the neutral, ion, and electron fluids, for the 15 generic types of reactions that are included in our network. ${\hat{G}}_{k}$, when multiplied by the appropriate reaction rate ${ \mathcal R }$ and summed over all of the chemical reactions in the network, gives the net component ${G}_{k}^{\mathrm{chem}}$, k = n, i, e.

Table 3. The Thermal Energy Source Terms ${\hat{G}}_{k}$, k = n, i, e Due to the Different Types of Chemical Reactions Included in the Model

Reaction type	Formula	${\hat{G}}_{n}$	${\hat{G}}_{i}$	${\hat{G}}_{e}$
Photodissociation [1]	A + h ν → B + C	ΔE	0	0
Photoionization [2]	A + h ν → A+ + e−	−Θn	Θn	ΔE
Photodissociative ionization [2],[3]	A + h ν → B + C+ + e−	$-\tfrac{{m}_{{\rm{C}}}}{{m}_{{\rm{A}}}}{{\rm{\Theta }}}_{n}$	$\tfrac{{m}_{{\rm{C}}}}{{m}_{{\rm{A}}}}{{\rm{\Theta }}}_{n}$	ΔE
	A + h ν → B + B' + C+ + e−	$-\tfrac{{m}_{{\rm{C}}}}{{m}_{{\rm{A}}}}{{\rm{\Theta }}}_{n}$	$\tfrac{{m}_{{\rm{C}}}}{{m}_{{\rm{A}}}}{{\rm{\Theta }}}_{n}$	ΔE
Neutral−neutral [2],[3]	A + B → C + D	ΔE	0	0
	A + B → C + D + E	ΔE	0	0
Ion−neutral a [1],[3]	A + B+ → C + D+	$\tfrac{{m}_{{\rm{A}}}{m}_{{\rm{D}}}+{m}_{{\rm{B}}}{m}_{{\rm{C}}}}{{m}_{{\rm{T}}}^{2}}({{\rm{\Theta }}}_{i}-{{\rm{\Theta }}}_{n})+\tfrac{{m}_{{\rm{D}}}}{{m}_{{\rm{T}}}}{\rm{\Delta }}E$	$\tfrac{{m}_{{\rm{A}}}{m}_{{\rm{D}}}+{m}_{{\rm{B}}}{m}_{{\rm{C}}}}{{m}_{{\rm{T}}}^{2}}({{\rm{\Theta }}}_{n}-{{\rm{\Theta }}}_{i})+\tfrac{{m}_{{\rm{C}}}}{{m}_{{\rm{T}}}}{\rm{\Delta }}E$	0
	A + B+ → C + ${\rm{C}}^{\prime}$ + D+	$\tfrac{{m}_{{\rm{A}}}{m}_{{\rm{D}}}+{m}_{{\rm{B}}}{m}_{\mathrm{CC}^{\prime} }}{{m}_{{\rm{T}}}^{2}}({{\rm{\Theta }}}_{i}-{{\rm{\Theta }}}_{n})+\tfrac{{m}_{{\rm{D}}}}{{m}_{{\rm{T}}}}{\rm{\Delta }}E$	$\tfrac{{m}_{{\rm{A}}}{m}_{{\rm{D}}}+{m}_{{\rm{B}}}{m}_{\mathrm{CC}^{\prime} }}{{m}_{{\rm{T}}}^{2}}({{\rm{\Theta }}}_{n}-{{\rm{\Theta }}}_{i})+\tfrac{{m}_{{\rm{C}}}}{{m}_{{\rm{T}}}}{\rm{\Delta }}E$	0
	A + B+ → C + ${\rm{C}}^{\prime}$ + C'' + D+	$\tfrac{{m}_{{\rm{A}}}{m}_{{\rm{D}}}+{m}_{{\rm{B}}}{m}_{\mathrm{CC}^{\prime} {\rm{C}}^{\prime\prime} }}{{m}_{{\rm{T}}}^{2}}({{\rm{\Theta }}}_{i}-{{\rm{\Theta }}}_{n})+\tfrac{{m}_{{\rm{D}}}}{{m}_{{\rm{T}}}}{\rm{\Delta }}E$	$\tfrac{{m}_{{\rm{A}}}{m}_{{\rm{D}}}+{m}_{{\rm{B}}}{m}_{\mathrm{CC}^{\prime} {\rm{C}}^{\prime\prime} }}{{m}_{{\rm{T}}}^{2}}({{\rm{\Theta }}}_{n}-{{\rm{\Theta }}}_{i})+\tfrac{{m}_{{\rm{C}}}}{{m}_{{\rm{T}}}}{\rm{\Delta }}E$	0
Radiative association of neutrals [2]	A + B → C + h ν	−Θn	0	0
Radiative association of ions & neutrals [2]	A + B+ → C+ + h ν	−Θn	$\tfrac{{m}_{{\rm{A}}}}{{m}_{{\rm{C}}}}({{\rm{\Theta }}}_{n}-{{\rm{\Theta }}}_{i})$	0
Ionizative association [3]	A + B → C+ + e−	−2Θn	2Θn	ΔE
Radiative de-excitation [3]	A* → A + h ν	0	0	0
Radiative recombination [1]	A+ + e− → A + h ν	Θi	−Θi	−Θe
Dissociative recombination [1],[3]	A+ + e− → B + C	Θi + Θe + ΔE	−Θi	−Θe
	A+ + e− → B + C + D	Θi + Θe + ΔE	−Θi	−Θe
	A+ + e− → B + C + D + E	Θi + Θe + ΔE	−Θi	−Θe
Electron impact ionization [2]	A + e− → A+ + e− + e−	−Θn	Θn	ΔE
Dissociative electron impact ionization [3]	A + e− → B + C+ + e− + e−	$-\tfrac{{m}_{{\rm{C}}}}{{m}_{{\rm{A}}}}{{\rm{\Theta }}}_{n}$	$\tfrac{{m}_{{\rm{C}}}}{{m}_{{\rm{A}}}}{{\rm{\Theta }}}_{n}$	ΔE
	A + e− → B + ${\rm{B}}^{\prime}$ + C+ + e− + e−	$-\tfrac{{m}_{{\rm{C}}}}{{m}_{{\rm{A}}}}{{\rm{\Theta }}}_{n}$	$\tfrac{{m}_{{\rm{C}}}}{{m}_{{\rm{A}}}}{{\rm{\Theta }}}_{n}$	ΔE
Electron impact dissociation [2]	A + e− → B + C	0	0	ΔE
	A + e− → B + C + D	0	0	ΔE
	A + e− → B + C + D + E	0	0	ΔE
Electron impact excitation [3]	A + e− → A* + e−	0	0	ΔE

Notes. Θ_k is the mean kinetic energy of a particle belonging to the fluid k such that ${{\rm{\Theta }}}_{n}=\tfrac{3}{2}{k}_{B}{T}_{n}$, ${{\rm{\Theta }}}_{i}=\tfrac{3}{2}{k}_{B}{T}_{i}$, and Θ_e = k_B T_e . ΔE is the mean exo/endo-thermicity of the chemical reaction.

^a Term m_T is the sum of the masses of the reactants, ${m}_{\mathrm{CC}^{\prime} }={m}_{{\rm{C}}}+{m}_{{\rm{C}}^{\prime} }$ and ${m}_{\mathrm{CC}^{\prime} {{\rm{C}}}^{{\prime \prime} }}={m}_{{\rm{C}}}+{m}_{{\rm{C}}^{\prime} }+{m}_{{\rm{C}}^{\prime\prime} }$.

References: [1] Draine (1986), [2] Rodgers & Charnley (2002), [3] Weiler (2006).

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A.4.2. Elastic Scattering

Elastic scattering occurs between each of the fluids, resulting in three types of scattering processes, namely ion−neutral, electron−neutral, and electron−ion, for which the rates of exchange of energy per unit volume are ${G}_{i,n}^{\mathrm{elas}}$, ${G}_{e,n}^{\mathrm{elas}}$, and ${G}_{e,i}^{\mathrm{elas}}$, respectively.

The ion−neutral elastic scattering is a special case of the ion−neutral reaction A + B⁺ → C + D⁺ given in Table 3, with m_A = m_C, m_B = m_D, and ΔE = 0. The most abundant neutral species in the coma are the parent molecules H₂O, CO, and CO₂, and the most abundant ions with which these do not undergo chemical reactions are H₃O⁺, NH₄ ⁺, CH₃OH⁺, ${\mathrm{CH}}_{3}{\mathrm{OH}}_{2}^{+}$, and HCNH⁺. Thus, ion−neutral elastic collisions can be modeled as "pseudo-reactions" of the kind

$$\begin{eqnarray}&&{\rm{A}}+{{\rm{B}}}^{+}\to {\rm{A}}+{{\rm{B}}}^{+}\end{eqnarray} \tag{ A19 }$$

where there is only an exchange of energy and no change in the number densities of A and B. The rates for these reactions are calculated using Equation (5), with the Arrhenius coefficients α_j ≈ 10⁻¹⁰ cm³ s⁻¹, β_j = 0.5, and γ_j = 0. The net change in energy for each collision is:

$$\begin{eqnarray}&&{\hat{G}}_{i,n}^{\mathrm{elas}}=\displaystyle \frac{2{m}_{n}{m}_{i}}{{({m}_{n}+{m}_{i})}^{2}}\left({{\rm{\Theta }}}_{i}-{{\rm{\Theta }}}_{n}\right),\end{eqnarray} \tag{ A20 }$$

where m_n and m_i are the respective masses of the neutral and ionic species participating in the scattering process. ${G}_{i,n}^{\mathrm{elas}}$ is found by multiplying the energy change given by Equation (A20) with the collision rate per unit volume, and summing over all ion−neutral elastic collision reactions.

Similar to the case of ion−neutral scattering, the elastic scattering between electrons and neutrals is dominated by scattering from H₂O, CO, and CO₂ molecules, such that

$$\begin{eqnarray}&&{G}_{e,n}^{\mathrm{elas}}\approx {G}_{e,n}^{\mathrm{elas},{{\rm{H}}}_{2}{\rm{O}}}+{G}_{e,n}^{\mathrm{elas},\mathrm{CO}}+{G}_{e,n}^{\mathrm{elas},{\mathrm{CO}}_{2}},\end{eqnarray} \tag{ A21 }$$

where ${G}_{e,n}^{\mathrm{elas},{{\rm{H}}}_{2}{\rm{O}}}$, ${G}_{e,n}^{\mathrm{elas},\mathrm{CO}}$, and ${G}_{e,n}^{\mathrm{elas},{\mathrm{CO}}_{2}}$ are the rates of exchange of energy per unit volume due to elastic scattering of electrons from H₂O, CO, and CO₂ molecules, respectively. The momentum transfer collision cross sections of electrons in these neutral gases has been compiled by Itikawa (2002, 2015) and Itikawa & Mason (2005), over a wide range of electron energies. To find the collision rate per unit volume of electrons with a neutral species, the relevant cross sections are averaged over a Maxwellian velocity distribution, and then multiplied by the number densities of electrons and the colliding neutral species. The average change in energy per collision ${\hat{G}}_{e,n}^{\mathrm{elas}}$ can be found from Equation (A20), by replacing the mass of the ion with the electron mass and Θ_i with Θ_e . ${G}_{e,n}^{\mathrm{elas}}$ is then calculated by multiplying ${\hat{G}}_{e,n}^{\mathrm{elas}}$ with the corresponding collision rate and then summing over the three neutral species.

Elastic scattering between ions and electrons due to Coulomb interactions results in the following rate of energy change per unit volume (Draine 1980):

$$\begin{eqnarray}&&\begin{array}{l}{G}_{e,i}^{\mathrm{elas}}=1.37\times {10}^{-42}\displaystyle \frac{{n}_{i}^{2}}{{\mu }_{i}}{T}_{e}^{-1.5}({T}_{i}-{T}_{e})\mathrm{ln}\\ \quad \times \left(1.24\times {10}^{4}\sqrt{\displaystyle \frac{{T}_{e}^{3}}{{n}_{i}}}\right)\quad (\mathrm{erg}\ {\mathrm{cm}}^{-3}{{\rm{s}}}^{-1}),\end{array}\end{eqnarray} \tag{ A22 }$$

where n_i is the total number density of all ions and μ_i is the mean ionic molecular mass.

A.4.3. Inelastic Scattering

For inelastic electron−neutral scattering, H₂O and CO are the only neutral species that are considered. The per unit volume rate of cooling of the electron fluid due to these types of collisions is thus given by

$$\begin{eqnarray}&&{G}_{e,n}^{\mathrm{inel}}\approx {G}_{e,n}^{\mathrm{inel},{{\rm{H}}}_{2}{\rm{O}}}+{G}_{e,n}^{\mathrm{inel},\mathrm{CO}}.\end{eqnarray} \tag{ A23 }$$

Inelastic collisions of H₂O molecules with electrons results in rotational and vibrational excitation of the molecules. The net cooling rate per unit volume of electrons due to inelastic scattering with H₂O is:

$$\begin{eqnarray}&&{G}_{e,n}^{\mathrm{inel},{{\rm{H}}}_{2}{\rm{O}}}={G}_{\mathrm{rot},{{\rm{H}}}_{2}{\rm{O}}}+{G}_{\nu 1,{{\rm{H}}}_{2}{\rm{O}}}+{G}_{\nu 2,{{\rm{H}}}_{2}{\rm{O}}}.\end{eqnarray} \tag{ A24 }$$

The first term on the right is the cooling due to rotational excitation and the second and third terms are due to vibrational excitations. The analytical expressions for these were derived by Cravens & Korosmezey (1986) and are given below.

$$\begin{eqnarray}&&\begin{array}{l}{G}_{\mathrm{rot},{{\rm{H}}}_{2}{\rm{O}}}=\left[a+b\mathrm{ln}\left(\displaystyle \frac{{T}_{e}}{{T}_{n}}\right)\right]\\ \quad \times ({T}_{e}-{T}_{n}){T}_{e}^{-5/4}{n}_{{{\rm{H}}}_{2}{\rm{O}}}{n}_{e}\quad (\mathrm{eV}\ {\mathrm{cm}}^{-3}{{\rm{s}}}^{-1}).\end{array}\end{eqnarray} \tag{ A25 }$$

$$\begin{eqnarray}&&\begin{array}{l}{G}_{\nu {\rm{j}},{{\rm{H}}}_{2}{\rm{O}}}=8.37\times {10}^{13}{W}_{j}{T}_{e}^{-3/2}\\ \quad \times \left[1-\exp \left\{{W}_{j}/{k}_{B}({T}_{e}^{-1}-{T}_{n}^{-1})\right\}\right]\\ \quad \times {I}_{j}{T}_{e}{n}_{{{\rm{H}}}_{2}{\rm{O}}}{n}_{e}\quad (\mathrm{eV}\ {\mathrm{cm}}^{-3}{{\rm{s}}}^{-1}).\end{array}\end{eqnarray} \tag{ A26 }$$

Terms ${n}_{{{\rm{H}}}_{2}{\rm{O}}}$ and n_e are the number densities of water molecules and electrons, j = 1 corresponds to the vibrational transition (000) → (010), and j = 2 stands for the sum of the transitions (000) → (100) and (000) → (001). The values of a, b, W_j , and I_j in the above expressions are given by Cravens & Korosmezey (1986).

Vibrational and electronic transitions of CO due to inelastic collisions with electrons are considered to calculate the term ${G}_{e,n}^{\mathrm{inel},\mathrm{CO}}$. The rotational excitation of CO is ignored, because the cross sections for these transitions due to inelastic collisions with electrons is less than that of H₂O by up to four orders of magnitude. The vibrational cooling rate per unit volume for 0 → j vibrational excitation for CO can be calculated in the manner given by Waite & Cravens (1981) for molecular hydrogen:

$$\begin{eqnarray}&&\begin{array}{l}{G}_{\nu j,\mathrm{CO}}={n}_{\mathrm{CO}}{n}_{e}{\int }_{0}^{\infty }L(E,{T}_{n}){{vf}}_{e}\\ \quad \times (E,{T}_{e}){dE}\quad (\mathrm{eV}\ {\mathrm{cm}}^{-3}{{\rm{s}}}^{-1}).\end{array}\end{eqnarray} \tag{ A27 }$$

Here, n_CO and n_e are CO and electron number densities, L(E, T_n ) is the loss function for vibrational excitation and de-excitation, $v={(2E/{m}_{e})}^{1/2}$ is the electron thermal velocity, and f_e (E, T_e ) is the normalized Maxwellian electron energy distribution function. The loss function for the transition 0 → j is

$$\begin{eqnarray}&&\begin{array}{l}{L}_{\nu j}(E,{T}_{n})={W}_{j}\{{\sigma }_{j}(E)-\exp \\ \quad \times (-{W}_{j}/{{kT}}_{n})\left(\displaystyle \frac{E+{W}_{j}}{E}\right){\sigma }_{j}(E+{W}_{j})\}\end{array}\end{eqnarray} \tag{ A28 }$$

where W_j is the threshold for transition j and T_n is the neutral temperature. Term σ_j (E) is the energy-dependent cross section for transition j, taken from Itikawa (2015). The transitions j = 1, 2, 3 are considered; higher transitions are ignored because of reduced cross sections.

The reactions for the electronic excitation of CO by electron impact and its subsequent de-excitation are shown in Table 4. These reactions are included in the network, and the energy change ${\hat{G}}_{k}$ is calculated as described in Section A.4.1.

Table 4. Reactions Showing the Excitation and De-excitation of the Electronic States of CO, Resulting in the Cooling of the Electron Fluid (Schmidt et al. 1988)

Electron impact excitation	Radiative de-excitation
CO + e− → CO(1Π) + e−	CO(1Π) → CO + h ν
CO + e− → CO(3Π) + e−	CO(3Π) → CO + h ν
CO + e− → CO(3Σ) + e−	CO(3Σ) → CO + h ν
CO + e− → CO(3Δ) + e−	CO(3Σ) → CO(3Π)+ h ν
	CO(3Δ) → CO + h ν
	CO(3Δ) → CO(3Π) + h ν

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Neutral molecules with a permanent dipole moment can be excited to higher rotational states due to self-collisions. Energy can be lost from the neutral fluid (mainly H₂O) due to infrared rotational emission, resulting in the cooling of the neutral fluid, such that

$$\begin{eqnarray}&&{G}_{n}^{\mathrm{rad}}\approx {G}_{n}^{\mathrm{rad},{{\rm{H}}}_{2}{\rm{O}}}.\end{eqnarray} \tag{ A29 }$$

An empirical formula for the energy loss due to H₂O–H₂O collisions was given by Shimizu (1976):

$$\begin{eqnarray}&&{G}_{n}^{\mathrm{rad},{{\rm{H}}}_{2}{\rm{O}}}=\displaystyle \frac{8.5\times {10}^{-19}{T}_{n}^{2}{n}_{{{\rm{H}}}_{2}{\rm{O}}}^{2}}{{n}_{{{\rm{H}}}_{2}{\rm{O}}}+2.7\times {10}^{7}{T}_{n}}\quad (\mathrm{erg}\ {\mathrm{cm}}^{-3}{{\rm{s}}}^{-1}).\end{eqnarray} \tag{ A30 }$$

Since optical depth effects cause radiation trapping in the inner regions of the coma, the radiation cooling rates are scaled by $\exp (-{\tau }_{\mathrm{IR}})$; (Schmidt et al. 1988), where

$$\begin{eqnarray}&&{\tau }_{\mathrm{IR}}(r)\approx 0.4\times n({r}_{0}){r}_{0}^{2}\displaystyle \frac{{\sigma }_{\mathrm{IR}}}{r}.\end{eqnarray} \tag{ A31 }$$

Here, r₀ is the nuclear radius, n is the number density of H₂O, and σ_IR = 4 × 10⁻¹⁵ cm² is the average infrared absorption cross section per molecule.

Solar UV radiation causes photodissociation, photoionization, and photodissociative ionization of the gaseous cometary species. The absorption of the UV radiation in the coma causes attenuation of the UV flux. If ϕ_i,∞ is the unattenuated flux corresponding to the wavelength λ_i , then the UV flux at a cometocentric distance r is given by

$$\begin{eqnarray}&&{\phi }_{i}(r)={\phi }_{i,\infty }{e}^{-{\tau }_{i}(r)},\end{eqnarray} \tag{ A32 }$$

where τ_i is the optical depth at wavelength λ_i , and can be calculated from the wavelength-dependent photochemical reaction cross sections and species number densities. For a cometary species denoted by j, if n_j is its number density and σ_ij,tot is its total photolytic cross section in λ_i , then the contribution of this species to the optical depth is:

$$\begin{eqnarray}&&{\tau }_{{ij}}(r)={\int }_{r}^{\infty }{\sigma }_{{ij},\mathrm{tot}}{n}_{j}(r){dr}.\end{eqnarray} \tag{ A33 }$$

The total optical depth at distance r can be obtained by summing over j, such that τ_i (r) = ∑_j τ_ij (r). Once the fluxes are known, the photolytic rate coefficient in the wavelength interval λ_i and λ_i + Δλ_i , and at a distance r can be found out using the relation

$$\begin{eqnarray}&&{k}_{i}(r)={\int }_{{\lambda }_{i}}^{{\lambda }_{i}+{\rm{\Delta }}{\lambda }_{i}}\sigma (\lambda )\phi (\lambda ,r)d\lambda .\end{eqnarray} \tag{ A34 }$$

This can be approximated as k_i (r) = σ_i Φ_i (r), where σ_i is the wavelength-averaged photo cross section in the ith bin that has a width of Δλ_i , and Φ_i (r) is the attenuated spectral photon flux integrated over the same wavelength bin. The total rate coefficient for any photolytic process can then be written as:

$$\begin{eqnarray}&&k(r)=\sum _{i}{k}_{i}(r).\end{eqnarray} \tag{ A35 }$$